**Adaptive Discretization Strategies for Parameter
Identification in Partial Differential Equations**

Parameter identification in partial differential equation is a class of large scale inverse problems demanding for highly efficient solution strategies. On the other hand, due to the inherent ill-posedness of such problems, any solution approach needs to contain regularization.

After a short motivation for the use of adaptivity in inverse problems for PDEs, we intend to give an at least partial overview on existing literature and report on our own research, which includes joint work with Hend Benameur, Anke Griesbaum, Alana Kirchner, Jonas Offtermatt, and Boris Vexler.

We will mainly dwell on two approaches. The first one is a method based on refinement and coarsening indicators where sensitivities of the data misfit functional with respect to changes in the discretization are computed as Largange multipliers of appropriately constrained misfit minimization problems. The second one originates from PDE constrained optimal control and relies on adaptive refinement according to goal oriented error estimators.

In both cases, special care has to be taken due to the ill-posedness of the underlying inverse problem in the sense that either stability has to be additionally incorporated or the stabilizing effect of discretization has to be approproately exploited. Therefore, key tasks in this context are on one hand to prove regularization properties for the resulting methods and on the other hand to show their efficiency for the solution of large scale inverse problems.

Aachen Institute for Advanced Study in Computational Engineering Science
(AICES) |